Magnetic Pole Equivalence and Performance Analyses of Multi-Layer Flux-Barrier Combined-Pole Permanent-Magnet Synchronous Machines Used for Electric Vehicles

Abstract

Multi-layer flux-barrier combined-pole permanent-magnet synchronous machines (MLFB-CP-PMSMs) are especially suitable for machines used in electrical vehicles (EVs), as they represent a tradeoff between electromagnetic performance and the consumption of high-priced rare-earth permanent magnets (PM). In this paper, magnetic pole equivalence and performance analyses of the MLFB-CP-PMSM are investigated. Firstly, three types of PM arrangements of combined poles are introduced, namely, parallel, series and series–parallel. Then, the magnetic circuit model and magnetic pole equivalence principle of MLFB-CP-PMSMs with different PM arrangements are analyzed. After that, the accuracy of the equivalence method is studied by comparing the machine electromagnetic performance before and after equivalence. Finally, the MLFB-CP-PMSM’s performance, including the loss, efficiency and electromagnetic torque, is analyzed. The results prove that the MLFB-CP-PMSM has the advantage of high efficiency, and the equivalence method can retain precision when the MLFB-CP-PMSM armature reaction degree varies.

1. Introduction

Global carbon dioxide emissions have increased rapidly in recent years, causing a series of climate problems [1,2,3]. Traditional vehicles use non-renewable fossil fuels to provide power and, at the same time, they produce large amounts of carbon dioxide emissions that are harmful to the environment. Hence, electric vehicles (EVs) are gradually replacing traditional vehicles due to their advantages of energy conservation and environmental protection [4,5,6]. The sales volume of EVs and hybrid electric vehicles (HEVs) is expected to exceed 48 million and thus exceed the sales volume of traditional vehicles by 2040 [7]. As the core component of EVs’ driving systems, the motor plays an important role in the performance of EVs [8]. The multi-layer flux-barrier combined-pole permanent-magnet synchronous machines (MLFB-CP-PMSMs) are appropriately used for EVs as driving motors based on a tradeoff between performance and the consumption of high-priced rare-earth permanent magnets (PM).

The multi-layer flux-barrier structure can improve reluctance torque and reduce PM usage. In [9], a magnetic circuit model of a multi-layer flux-barrier permanent-magnet synchronous machine (PMSM) is established. The optimal machine scheme with the least PM usage was obtained by taking the anti-demagnetization ability and output torque into consideration. The torque harmonics of a multi-layer flux-barrier PMSM are analyzed and the method of reducing the torque ripple is studied in [10,11]. The PM working point of a multi-layer flux-barrier PMSM is calculated in [12], and the machine’s anti-demagnetization ability is analyzed.

The combined-pole structure, specifically that using a variety of PM materials in the magnetic pole, can reduce the consumption of a single PM material and integrate the advantages of various PM materials. In [13], the air gap flux density distribution of a surface-mounted PMSM is analyzed, and a “pyramid” combined-pole surface-mounted PMSM is proposed, of which the back electromotive force (EMF) is close to a sine wave. In [14], the multi-objective optimization of the combined-pole interior PMSM is studied to improve its performance and reliability. Three combined-pole interior PMSM topologies are proposed in [15], and their electromagnetic performances are analyzed and compared. It was found that the combined-pole interior PMSM has a better anti-demagnetization ability at high temperatures than the conventional single-PM-material interior PMSM, because the ferrite has a positive temperature coefficient of intrinsic coercivity. This means that the MLFB-CP-PMSMs are especially suitable for high-electric- and high-magnetic-load applications, in which the machine temperature rise is high and the NdFeB in the machine can easily demagnetize.

The MLFB-CP-PMSM integrates the merits of multi-layer flux-barrier and combined-pole structure [16,17,18,19]. In [20], the reluctance torque of the MLFB-CP-PMSM is obtained, and it is found that the current angles corresponding to the maximum reluctance torque and maximum electromagnetic torque are approximate. The influences of the PM thickness and pole arc coefficient on the MLFB-CP-PMSM’s back electromotive force (EMF) are investigated in [21], and the optimal structure parameters of machine are selected. The MLFB-CP-PMSM’s anti-demagnetization capability at different temperatures is analyzed in [22,23]. The results show that the MLFB-CP-PMSM has high anti-demagnetization capability at high temperatures, but the ferrite PMs are prone to irreversible demagnetization when the temperature is low. Variations in MLFB-CP-PMSM inductance and the saliency ratio with the air gap length are studied in [24]. In [25], a novel MLFB-CP-PMSM is proposed, and the torque versus the current amplitude is investigated. It is found that the proposed MLFB-CP-PMSM has a good overload capability. The influence of temperature on MLFB-CP-PMSM electromagnetic performance is researched in [26]. The results show that the torque decreases and that the torque ripple is nearly unchanged when the temperature increases. In [27], the equivalent magnetic circuit model of the MLFB-CP-PMSM is established, but the theoretical analysis based on the magnetic circuit model is still inadequate.

The combined-pole structure can create a tradeoff between the electromagnetic performance and usage of high-priced rare-earth PM, but it also causes the MLFB-CP-PMSM to have a complex structure and is difficult to analyze. Various PM materials in the MLFB-CP-PMSM can be equivalent to one PM material, which will reduce the structural complexity and difficulty of the theoretical analysis of the MLFB-CP-PMSM. However, this topic has not been studied in the literature.

In this paper, the equivalent magnetic circuit models of MLFB-CP-PMSMs adopting series, parallel and series–parallel PM arrangements are established. According to the magnetic pole equivalence principle, the relationship between the PM property parameters and PM structure parameters before and after the equivalence is derived. Finite element analysis (FEA) is applied to verify the equivalence method’s effectiveness. Finally, the loss, efficiency and torque characteristics of the MLFB-CP-PMSMs adopting three types of PM arrangements are studied and compared.

2. Parallel Magnetic Pole Equivalence

The magnetic pole equivalence method for the MLFB-CP-PMSM is analyzed in this section, which means that various PM materials in the same flux barrier are equivalent to one PM material. The equivalent MLFB-CP-PMSMs are more convenient for theoretical analysis and electromagnetic calculation.

According to the PM arrangements, the magnetic pole of the MLFB-CP-PMSM can be divided into three types, namely, parallel, series and series–parallel. Figure 1 shows the PM arrangements of three magnetic pole types. In Figure 1, the arrows are the PM magnetization direction, and the different colors represent different PM materials. As shown in Figure 1, the PMs of the parallel type are arranged left and right in the magnetization direction; the PMs of the series type are arranged up and down in the magnetization direction; and the PMs in the series–parallel type are arranged up and down, as well as left and right, in the magnetization direction. The machine structure is the same except for the PM structure when the MLFB-CP-PMSMs adopting different magnetic pole types are analyzed.

Figure 2a shows the MLFB-CP-PMSM structure with a parallel PM arrangement. Figure 2b shows sketches of the flux barriers, bridges and PMs. The PMs in the MLFB-CP-PMSM include the NdFeB and ferrite. The main dimensions and parameters of the investigated MLFB-CP-PMSM are chosen according to the motor requirements used for EVs.

Table 1. Main dimensions and parameters of the investigated MLFB-CP-PMSM.

Items	Unit	Value
Stator outer diameter	mm	290
Rotor outer diameter	mm	172
Air gap	mm	0.8
Stack length	mm	75
Rated current	A	100
Maximum speed	rpm	7000
Maximum efficiency	%	94

shows the main motor dimensions and parameters.

In this paper, the basic principle of magnetic pole equivalence includes the following conditions: (1) various PMs in the same flux barrier are equivalent to one PM material on the basis of keeping the machine’s electromagnetic performance unchanged, i.e., keeping the machine’s magnetic field distribution unchanged; and (2) the magnetic parameters of the PMs (including the remanence and relative permeability) are changed after equivalence, but the PM structure parameters remain the same before and after equivalence.

2.1. Equivalent Principle Analysis

Figure 3 shows the magnetic circuit model of the MLFB-CP-PMSM with a parallel PM arrangement, and it is rewritten in Figure 4. In this paper, only the magnetic circuit model of the half pole is established, as the magnetic circuit is symmetrical. The silicon steel reluctance is ignored due to the high permeability of silicon steel.

In Figure 3 and Figure 4, F_i is the magnetomotive force (MMF) produced by the stator current; R_Bai and R_gi are the flux barrier reluctance and air gap reluctance, respectively; R_{Bri_Pi} is the reluctance of the saturated bridge; F_{P_(2i−1)} and R_{P_(2i−1)} are the equivalent MMF and reluctance of the NdFeB in the i-layer flux barrier; and F_{P_(2i)} and R_{P_(2i)} are the equivalent MMF and reluctance of the ferrite in the i-layer flux barrier. F_{P_(2i−1)}, R_{P_(2i−1)}, F_{P_(2i)} and R_{P_(2i)} can be expressed as:

$$F_{P\_(2i-1)}=\frac{B_{r1}}{{\mu }_0{\mu }_{r1}}{h}_{P\_(2i-1)}$$

(1)

$$R_{P\_(2i-1)}=\frac{h_{P\_(2i-1)}}{{\mu }_0{\mu }_{r1}{b}_{P\_(2i-1)}L}$$

(2)

$$F_{P\_(2i)}=\frac{B_{r2}}{{\mu }_0{\mu }_{r2}}{h}_{P\_(2i)}$$

(3)

$$R_{P\_(2i)}=\frac{h_{P\_(2i)}}{{\mu }_0{\mu }_{r2}{b}_{P\_(2i)}L}$$

(4)

where B_r1 and μ_r1 are the remanence and relative permeability of NdFeB; B_r2 and μ_r2 are the remanence and relative permeability of ferrite; h_{P_(2i−1)} and b_{P_(2i−1)} are the thickness and width of the NdFeB in the i-layer flux barrier, as shown in Figure 5; h_{P_(2i)} and b_{P_(2i)} are the thickness and width of the ferrite in the i-layer flux barrier, as shown in Figure 5; μ₀ is the vacuum permeability; and L is the stack length.

Table 2. Values of h_{P_(2i−1)}, b_{P_(2i−1)}, h_{P_(2i)} and b_{P_(2i)}.

Items	Unit	Value
hP_1, hP_2	mm	4
hP_3, hP_4	mm	4
hP_5, hP_6	mm	4
bP_1	mm	3
bP_3	mm	8
bP_5	mm	13
bP_2, bP_4, bP_6	mm	2

shows the values of h_{P_(2i−1)}, b_{P_(2i−1)}, h_{P_(2i)} and b_{P_(2i)}.

Figure 6 shows the magnetic circuit model of the MLFB-CP-PMSM with a parallel PM arrangement after equivalence, and it is rewritten in Figure 7. In Figure 6 and Figure 7, R′_{Bri_Pi} is the reluctance of the saturated bridge after equivalence; F′_{P_i} and R′_{P_i} are the equivalent MMF and reluctance of the PMs after equivalence. F′_{P_i} and R′_{P_i} can be expressed as:

$${F^{\prime }}_{P\_(2i-1)}=\frac{B_{r\_Pi}}{{\mu }_0{\mu }_{r\_Pi}}{h}_{P\_(2i-1)}$$

(5)

$${R^{\prime }}_{P\_(2i-1)}=\frac{h_{P\_(2i-1)}}{{\mu }_0{\mu }_{r\_Pi}{b}_{P\_(2i-1)}L}$$

(6)

$${F^{\prime }}_{P\_(2i)}=\frac{B_{r\_Pi}}{{\mu }_0{\mu }_{r\_Pi}}{h}_{P\_(2i)}$$

(7)

$${R^{\prime }}_{P\_(2i)}=\frac{h_{P\_(2i)}}{{\mu }_0{\mu }_{r\_Pi}{b}_{P\_(2i)}L}$$

(8)

where B_{r_Pi} and μ_{r_Pi} are the remanence and relative permeability of the PMs after equivalence in the i-layer flux barrier.

According to the magnetic pole equivalence principle of keeping the machine’s magnetic field distribution unchanged, we know that the reluctance of the saturated bridge remains unchanged before and after equivalence. By comparing Figure 4 and Figure 7, it is found that only the equivalent MMF and reluctance of the PMs change after equivalence. Hence, the flux produced by all the PMs and the reluctance of all the PMs in the i-layer flux barrier should be unchanged before and after equivalence according to the equivalence principle. Therefore, the following equations are obtained:

$$\frac{F_{P\_(2i-1)}}{R_{P\_(2i-1)}}+\frac{F_{P\_(2i)}}{R_{P\_(2i)}}=\frac{{F^{\prime }}_{P\_(2i-1)}}{{R^{\prime }}_{P\_(2i-1)}}+\frac{{F^{\prime }}_{P\_(2i)}}{{R^{\prime }}_{P\_(2i)}}$$

(9)

$$\frac{1}{R_{P\_(2i-1)}}+\frac{1}{R_{P\_(2i)}}=\frac{1}{{R^{\prime }}_{P\_(2i-1)}}+\frac{1}{{R^{\prime }}_{P\_(2i)}}$$

(10)

According to Equations (1)–(10), the following equations can be obtained:

$$B_{r\_Pi}=\frac{B_{r1}{b}_{P\_(2i-1)}+B_{r2}{b}_{P\_(2i)}}{b_{P\_(2i-1)}+b_{P\_(2i)}}$$

(11)

$${\mu }_{r\_Pi}=\frac{{\mu }_{r1}{h}_{P\_(2i)}{b}_{P\_(2i-1)}+{\mu }_{r2}{h}_{P\_(2i-1)}{b}_{P\_(2i)}}{h_{P\_(2i)}{b}_{P\_(2i-1)}+h_{P\_(2i-1)}{b}_{P\_(2i)}}$$

(12)

2.2. Equivalent Result Comparisons

The electromagnetic performances of the machines with a parallel PM arrangement before and after equivalence are compared to verify the equivalence method’s accuracy. Figure 8 and Figure 9 show the no-load back electromotive force (EMF) and electromagnetic torque at the rated current obtained through the FEA of the MLFB-CP-PMSMs with a parallel PM arrangement.

As found in Figure 8 and Figure 9, the waveforms of no-load back EMF and electromagnetic torque before and after equivalence match well. The fundamental amplitude of no-load back EMF and the constant component of electromagnetic torque before and after equivalence have the errors of 0.04% and 1.04%, which can be considered negligible.

3. Series Magnetic Pole Equivalence

Figure 10a shows the MLFB-CP-PMSM structure with a series PM arrangement, and Figure 10b shows a sketch of the rotor details. The PMs in the MLFB-CP-PMSM with a series PM arrangement are NdFeB and ferrite.

3.1. Equivalent Principle Analysis

Figure 11 shows the magnetic circuit model of the MLFB-CP-PMSM with a series PM arrangement, and it is rewritten in Figure 12. In Figure 11 and Figure 12, R_{Bri_Si} is the reluctance of the saturated bridge; F_{S_(2i−1)} and R_{S_(2i−1)} are the equivalent MMF and reluctance of the ferrite in the i-layer flux barrier; and F_{S_(2i)} and R_{S_(2i)} are the equivalent MMF and reluctance of the NdFeB in the i-layer flux barrier.

F_{S_(2i−1)}, R_{S_(2i−1)}, F_{S_(2i)} and R_{S_(2i)} can be expressed as:

$$F_{S\_(2i-1)}=\frac{B_{r2}}{{\mu }_0{\mu }_{r2}}{h}_{S\_(2i-1)}$$

(13)

$$R_{S\_(2i-1)}=\frac{h_{S\_(2i-1)}}{{\mu }_0{\mu }_{r2}{b}_{S\_(2i-1)}L}$$

(14)

$$F_{S\_(2i)}=\frac{B_{r1}}{{\mu }_0{\mu }_{r1}}{h}_{S\_(2i)}$$

(15)

$$R_{S\_(2i)}=\frac{h_{S\_(2i)}}{{\mu }_0{\mu }_{r1}{b}_{S\_(2i)}L}$$

(16)

where h_{S_(2i−1)} and b_{S_(2i−1)} are the thickness and width of the ferrite in the i-layer flux barrier, as shown in Figure 13; h_{S_(2i)} and b_{S_(2i)} are the thickness and width of the NdFeB in the i-layer flux barrier, as shown in Figure 13.

Table 3. Values of h_{S_(2i−1)}, b_{S_(2i−1)}, h_{S_(2i)} and b_{S_(2i)}.

Items	Unit	Value
hS_1	mm	1.6
hS_2	mm	2.4
hS_3	mm	0.8
hS_4	mm	3.2
hS_5	mm	0.53
hS_6	mm	3.47
bS_1, bS_2	mm	5
bS_3, bP_4	mm	10
bS_5, bS_6	mm	15

shows the values of h_{S_(2i−1)}, b_{S_(2i−1)}, h_{S_(2i)} and b_{S_(2i)}. h_{S_3} and h_{S_5} are small, which means that the corresponding PMs are thin. The method of dividing the PM into blocks in the axial direction can be used to manufacture a thin PM and ensure that the PM size meets the requirements.

Figure 14 shows the magnetic circuit model of the MLFB-CP-PMSM with a series PM arrangement after equivalence, and it is rewritten in Figure 15.

In Figure 14 and Figure 15, R′_{Bri_Si} is the reluctance of the saturated bridge after equivalence; F′_{S_i} and R′_{S_i} are the equivalent MMF and reluctance of the PMs after equivalence. F′_{S_i} and R′_{S_i} are calculated as:

$${F^{\prime }}_{S\_(2i-1)}=\frac{B_{r\_Si}}{{\mu }_0{\mu }_{r\_Si}}{h}_{S\_(2i-1)}$$

(17)

$${R^{\prime }}_{S\_(2i-1)}=\frac{h_{S\_(2i-1)}}{{\mu }_0{\mu }_{r\_Si}{b}_{S\_(2i-1)}L}$$

(18)

$${F^{\prime }}_{S\_(2i)}=\frac{B_{r\_Si}}{{\mu }_0{\mu }_{r\_Si}}{h}_{S\_(2i)}$$

(19)

$${R^{\prime }}_{S\_(2i)}=\frac{h_{S\_(2i)}}{{\mu }_0{\mu }_{r\_Si}{b}_{S\_(2i)}L}$$

(20)

where B_{r_Si} and μ_{r_Si} are the remanence and relative permeability of the PMs after equivalence in the i-layer flux barrier.

According to the equivalence principle, the following equations can be obtained on the basis of keeping the magnetic field distribution unchanged before and after equivalence:

$$F_{S\_(2i-1)}+F_{S\_(2i)}={F^{\prime }}_{S\_(2i-1)}+{F^{\prime }}_{S\_(2i)}$$

(21)

$$R_{S\_(2i-1)}+R_{S\_(2i)}={R^{\prime }}_{S\_(2i-1)}+{R^{\prime }}_{S\_(2i)}$$

(22)

According to Equations (13)–(22), the following equations are obtained:

$$B_{r\_Si}=\frac{(B_{r1}{h}_{S\_(2i)}{\mu }_{r2}+B_{r2}{h}_{S\_(2i-1)}{\mu }_{r1})(h_{S\_(2i)}{b}_{S\_(2i-1)}+h_{S\_(2i-1)}{b}_{S\_(2i)})}{(h_{S\_(2i)}+h_{S\_(2i-1)})(h_{S\_(2i)}{b}_{S\_(2i-1)}{\mu }_{r2}+h_{S\_(2i-1)}{b}_{S\_(2i)}{\mu }_{r1})}$$

(23)

$${\mu }_{r\_Si}=\frac{(h_{S\_(2i)}{b}_{S\_(2i-1)}+h_{S\_(2i-1)}{b}_{S\_(2i)}){\mu }_{r1}{\mu }_{r2}}{h_{S\_(2i)}{b}_{S\_(2i-1)}{\mu }_{r2}+h_{S\_(2i-1)}{b}_{S\_(2i)}{\mu }_{r1}}$$

(24)

3.2. Equivalent Result Comparisons

In this section, the electromagnetic performances of the MLFB-CP-PMSM with a series PM arrangement before and after equivalence are compared. Figure 16 and Figure 17 show the no-load back EMF and electromagnetic torque at the rated current of the MLFB-CP-PMSM with a series PM arrangement. As seen in Figure 16 and Figure 17, the fundamental amplitude of no-load back EMF and the constant component of electromagnetic torque before and after equivalence have the errors of 0.94% and 0.3%, respectively.

4. Series–Parallel Magnetic Pole Equivalence

The MLFB-CP-PMSM with a series–parallel PM arrangement is shown in Figure 18a. A sketch of the rotor details is shown in Figure 18b. Except for the PM configuration, the machine structure and PM materials in Figure 18 are the same as those of the MLFB-CP-PMSM with series and parallel PM arrangements.

4.1. Equivalent Principle Analysis

Figure 19 shows the magnetic circuit model of the MLFB-CP-PMSM with a series–parallel PM arrangement, and it is rewritten in Figure 20. In Figure 19 and Figure 20, R_{Bri_SPi} is the reluctance of the saturated bridge; F_{SP_(3i-1)} and R_{SP_(3i-1)} are the equivalent MMF and reluctance of the ferrite in the i-layer flux barrier; and F_{SP_(3i-2)}, R_{SP_(3i-2)}, F_{SP_(3i)} and R_{SP_(3i)} are the equivalent MMF and reluctance of the NdFeB in the i-layer flux barrier. The calculation formulae of F_{SP_(3i-2)}, R_{SP_(3i-2)}, F_{SP_(3i-1)}, R_{SP_(3i-1)}, F_{SP_(3i)} and R_{SP_(3i)} are as follows:

$$F_{SP\_(3i-2)}=\frac{B_{r1}}{{\mu }_0{\mu }_{r1}}{h}_{SP\_(3i-2)}$$

(25)

$$R_{SP\_(3i-2)}=\frac{h_{SP\_(3i-2)}}{{\mu }_0{\mu }_{r1}{b}_{SP\_(3i-2)}L}$$

(26)

$$F_{SP\_(3i-1)}=\frac{B_{r2}}{{\mu }_0{\mu }_{r2}}{h}_{SP\_(3i-1)}$$

(27)

$$R_{SP\_(3i-1)}=\frac{h_{SP\_(3i-1)}}{{\mu }_0{\mu }_{r2}{b}_{SP\_(3i-1)}L}$$

(28)

$$F_{SP\_(3i)}=\frac{B_{r1}}{{\mu }_0{\mu }_{r1}}{h}_{SP\_(3i)}$$

(29)

$$R_{SP\_(3i)}=\frac{h_{SP\_(3i)}}{{\mu }_0{\mu }_{r1}{b}_{SP\_(3i)}L}$$

(30)

where h_{SP_(3i-2)}, h_{SP_(3i-2)}, b_{SP_(3i)} and b_{SP_(3i)} are the thickness and width of the NdFeB in the i-layer flux barrier, as shown in Figure 21; h_{SP_(3i-1)} and b_{SP_(3i-1)} are the thickness and width of the ferrite in the i-layer flux barrier, as shown in Figure 21.

Table 4. Values of h_{SP_(3i-2)}, h_{SP_(3i-1)}, h_{SP_(3i)}, b_{SP_(3i-2)}, b_{SP_(3i-1)} and b_{SP_(3i)}.

Items	Unit	Value
hSP_1, hSP_4, hSP_7	mm	4
hSP_2, hSP_3	mm	2
hSP_5, hSP_6	mm	2
hSP_8, hSP_9	mm	2
bSP_1	mm	1
bSP_2, bSP_3	mm	4
bSP_4	mm	6
bSP_5, bSP_6	mm	4
bSP_7	mm	11
bSP_8, bSP_9	mm	4

shows the values of h_{SP_(3i-2)}, h_{SP_(3i-1)}, h_{SP_(3i)}, b_{SP_(3i-2)}, b_{SP_(3i-1)} and b_{SP_(3i)}.

Figure 22 shows the magnetic circuit model of the MLFB-CP-PMSM with a series–parallel PM arrangement after equivalence, and it is rewritten in Figure 23. In Figure 22 and Figure 23, R′_{Bri_SPi} is the reluctance of the saturated bridge after equivalence; F′_{SP_i} and R′_{SP_i} are the equivalent MMF and reluctance of the PMs after equivalence.

F′_{SP_i} and R′_{SP_i} can be calculated as follows:

$${F^{\prime }}_{SP\_(3i-2)}=\frac{B_{r\_SPi}}{{\mu }_0{\mu }_{r\_SPi}}{h}_{SP\_(3i-2)}$$

(31)

$${R^{\prime }}_{SP\_(3i-2)}=\frac{h_{SP\_(3i-2)}}{{\mu }_0{\mu }_{r\_SPi}{b}_{SP\_(3i-2)}L}$$

(32)

$${F^{\prime }}_{SP\_(3i-1)}=\frac{B_{r\_SPi}}{{\mu }_0{\mu }_{r\_SPi}}{h}_{SP\_(3i-1)}$$

(33)

$${R^{\prime }}_{SP\_(3i-1)}=\frac{h_{SP\_(3i-1)}}{{\mu }_0{\mu }_{r\_SPi}{b}_{SP\_(3i-1)}L}$$

(34)

$${F^{\prime }}_{SP\_(3i)}=\frac{B_{r\_SPi}}{{\mu }_0{\mu }_{r\_SPi}}{h}_{SP\_(3i)}$$

(35)

$${R^{\prime }}_{SP\_(3i)}=\frac{h_{SP\_(3i)}}{{\mu }_0{\mu }_{r\_SPi}{b}_{SP\_(3i)}L}$$

(36)

where B_{r_SPi} and μ_{r_SPi} are the remanence and relative permeability of the PMs after equivalence in the i-layer flux barrier.

On the basis of the equivalence principles, the following equations are obtained, keeping the magnetic field distribution unchanged before and after equivalence:

$$\frac{F_{SP\_(3i-2)}}{R_{SP\_(3i-2)}}+\frac{F_{SP\_(3i-1)}+F_{SP\_(3i)}}{R_{SP\_(3i-1)}+R_{SP\_(3i)}}=\frac{{F^{\prime }}_{SP\_(3i-2)}}{{R^{\prime }}_{SP\_(3i-2)}}+\frac{{F^{\prime }}_{SP\_(3i-1)}+{F^{\prime }}_{SP\_(3i)}}{{R^{\prime }}_{SP\_(3i-1)}+{R^{\prime }}_{SP\_(3i)}}$$

(37)

$$\frac{1}{R_{SP\_(3i-2)}}+\frac{1}{R_{SP\_(3i-1)}+R_{SP\_(3i)}}=\frac{1}{{R^{\prime }}_{SP\_(3i-2)}}+\frac{1}{{R^{\prime }}_{SP\_(3i-1)}+{R^{\prime }}_{SP\_(3i)}}$$

(38)

According to Equations (25)–(38), the following equations can be obtained:

$$B_{r\_SPi}=\frac{X(h_{SP\_(3i-1)}{b}_{SP\_(3i)}+h_{SP\_(3i)}{b}_{SP\_(3i-1)})}{{\mu }_{r1}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+{\mu }_{r2}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}}$$

(39)

$${\mu }_{r\_SPi}=\frac{Y(h_{SP\_(3i-1)}{b}_{SP\_(3i)}+h_{SP\_(3i)}{b}_{SP\_(3i-1)})}{{\mu }_{r1}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+{\mu }_{r2}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}}$$

(40)

where X and Y are expressed as:

$$X=\frac{B_{r1}{\mu }_{r1}{b}_{SP\_(3i-2)}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+B_{r1}{\mu }_{r2}{b}_{SP\_(3i-2)}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}+B_{r2}{\mu }_{r1}{h}_{SP\_(3i-1)}{b}_{SP\_(3i-1)}{b}_{SP\_(3i)}+B_{r1}{\mu }_{r2}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}{b}_{SP\_(3i)}}{b_{SP\_(3i-2)}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+b_{SP\_(3i-2)}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}+b_{SP\_(3i-1)}{b}_{SP\_(3i)}{h}_{SP\_(3i-1)}+b_{SP\_(3i-1)}{b}_{SP\_(3i)}{h}_{SP\_(3i)}}$$

(41)

$$Y=\frac{{\mu }_{r1}{\mu }_{r1}{b}_{SP\_(3i-2)}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+{\mu }_{r1}{\mu }_{r2}{b}_{SP\_(3i-2)}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}+{\mu }_{r1}{\mu }_{r2}{h}_{SP\_(3i-2)}{b}_{SP\_(3i-1)}{b}_{SP\_(3i)}}{b_{SP\_(3i-2)}{h}_{SP\_(3i-1)}{b}_{SP\_(3i)}+b_{SP\_(3i-2)}{h}_{SP\_(3i)}{b}_{SP\_(3i-1)}+h_{SP\_(3i-2)}{b}_{SP\_(3i-1)}{b}_{SP\_(3i)}}$$

(42)

4.2. Equivalent Result Comparisons

The electromagnetic performances of the MLFB-CP-PMSM with a series–parallel PM arrangement before and after equivalence are compared in this section. Figure 24 and Figure 25 show the no-load back EMF and electromagnetic torque at the rated current of the MLFB-CP-PMSM with a series–parallel PM arrangement. The fundamental amplitude of no-load back EMF and the constant component of electromagnetic torque before and after equivalence have the errors of 1.6% and 1.2%. Moreover, the waveforms of no-load back EMF and electromagnetic torque before and after equivalence are in good agreement.

5. Performance Analyses

This section analyzes the loss, efficiency and electromagnetic torque characteristics of the MLFB-CP-PMSMs, which are vital for the performance of the machines used for EVs.

5.1. Loss and Efficiency Analyses

The stator core loss and rotor core loss when the MLFB-CP-PMSMs adopting different PM arrangements operate at the rated current and rated speed (2000 rpm) are shown in Figure 26. As shown in Figure 26, the stator core loss is larger than the rotor loss in the MLFB-CP-PMSMs adopting three different PM arrangements. This is because the speed of rotor is the same as that of the fundamental magnetic field, which results in the reduction in rotor core loss. Moreover, it is known that the stator core loss and rotor core loss of the equivalent machine are close to those of the machine before equivalence.

The PMs’ eddy current loss, copper loss, core loss and mechanical loss and the efficiency of the MLFB-CP-PMSMs with the three PM arrangements are given in

Table 5. Loss and efficiency of MLFB-CP-PMSM.

Items	Parallel PM	Series PM	Series–Parallel PM
PMs’ eddy current loss (W)	2.3	4.1	1.6
Copper loss (W)	1061.7	1061.7	1061.7
Rotor core loss (W)	15.1	15.9	15.5
Stator core loss (W)	140.4	143.5	141.7
Mechanical loss (W)	11.6	11.6	11.6
Efficiency (%)	94.8	94.8	94.8

. The mechanical loss p_fw is calculated using the following equation:

$$p_{fw}=16p{(\frac{v}{40})}^3\sqrt{\frac{L}{19}\times {10}^3}$$

(43)

where p is the machine pole pairs; v is the circumferential speed of the rotor.

As shown in Table 5, the PMs’ eddy current loss is negligible compared with the core loss and copper loss. The stator winding is applied with the same rated current in all three MLFB-CP-PMSMs; hence, the copper loss is the also same. The efficiencies of all three MLFB-CP-PMSMs reach 94.8%, which means that the MLFB-CP-PMSMs have the advantage of high efficiency and meet the high efficiency requirements of the driving motors used for EVs.

5.2. Electromagnetic Torque Analysis

Figure 27 shows the average electromagnetic torque of the MLFB-CP-PMSMs with the three PM arrangements applying different winding currents. In Figure 27, I_N is the rated current.

When the winding current changes, the results obtained before and after equivalence match well, which means that the equivalence method retains its precision as the MLFB-CP-PMSM armature reaction degree varies. The electromagnetic torque consists of reluctance torque and PM torque. The reluctance torque and PM torque increase with the increment in the winding current; hence, the electromagnetic torque increases when the winding current increases.

The three types of PM arrangements of combined poles use the relatively small amount ferrite to replace the NdFeB in order to avoid a significant decrease in machine performance. The ferrite volume for the three types of PM arrangements is also the same. Hence, the electromagnetic torque and loss of the MLFB-CP-PMSMs adopting the three types of PM arrangements have a small difference, as can be seen in Figure 26 and Figure 27 and Table 5.

5.3. Demagnetization Analysis

The ferrite and NdFeB temperature coefficients of intrinsic coercivity are positive and negative, respectively. This means that the ferrite is prone to demagnetization at low temperatures, but the NdFeB is prone to demagnetization at high temperatures. Hence, the demagnetization curve temperatures of the ferrite and NdFeB are selected as 20 °C and 100 °C in this section. In this paper, the NdFeB and ferrite grades are N35UH and DM4545. Their demagnetization curves at different temperatures are shown in Figure 28. As can be seen in Figure 28, the knee points of N35UH at 100 °C and DM4545 at 20 °C are below 0 T. This means that if the flux densities of the ferrite and NdFeB are not less than 0 T, i.e., the PM’s flux density is consistent with the PM’s magnetization direction, the ferrite and NdFeB will not demagnetize. Moreover, when the machine’s demagnetization is studied, the stator current is selected as the maximum current (1.5 I_N), and the stator magnetomotive force (MMF) is completely against the PM magnetization direction.

The PMs’ flux density distributions are shown in Figure 29 when the demagnetization current is applied to the MLFB-CP-PMSMs with the three PM arrangements. It can be found that the PMs’ flux density is more than 0 T in the MLFB-CP-PMSMs with the three PM arrangements, which proves that the PMs in the machine will not demagnetize.

5.4. Structural Analysis

This paper investigates the structural analysis of the machine rotor to avoid mechanical damage of the rotor structure caused by centrifugal force when the machine operates at high speed. The rotor stress and deformation distributions of MLFB-CP-PMSMs with three PM arrangements at maximum speed are shown in Figure 30, Figure 31 and Figure 32. The ultimate tensile strengths of the ferrite and silicon steel are approximately 30 MPa and 500 MPa, respectively. It can be seen that the maximum stresses of the ferrite and silicon steel are lower than their respective ultimate tensile strengths. Hence, the centrifugal force cannot cause mechanical damage of the MLFB-CP-PMSMs’ rotors.

Moreover, it can be seen that the PMs’ deformation is very small in Figure 30, Figure 31 and Figure 32; hence, the influence of centrifugal force on the bonding performance between the NdFeB and ferrite is also very small.

6. Conclusions

This paper proposed three PM arrangements of MLFB-CP-PMSMs used for EVs, namely, the parallel type, series type, series–parallel type. The magnetic circuit models of the different MLFB-CP-PMSMs were determined, and the magnetic pole equivalence method was studied. The relationships of the PM magnetic parameters after equivalence with the PM structure parameters and PM magnetic parameters before equivalence were deduced. The errors in machine performance before and after equivalence are not more than 1.6%, which proves that the equivalence method has a high accuracy. The equivalence method can also be extended to MLFB-CP-PMSMs with more magnetic pole materials and more complex magnetic pole structures. Moreover, the MLFB-CP-PMSMs’ performances were studied. The results show that the efficiencies of all three MLFB-CP-PMSMs reach 94.8%, which means that the MLFB-CP-PMSMs can meet the high efficiency requirements of the driving motors used for EVs. The electromagnetic torque increases as the winding current increases, and the equivalence method still retains a high accuracy when the MLFB-CP-PMSM armature reaction degree changes. Corresponding experimental research on the MLFB-CP-PMSMs will be carried out in future work.